Consider a two-dimensional lattice in the \(x-y\) plane with sides of length \(L_{x}\) and \(L_{y}\) which contains \(N\) atoms ( \(N\) very large) coupled by nearest-neighbor harmonic forces. (a) Compute the Debye frequency for this lattice. (b) In the limit \(T \rightarrow 0\), what is the heat capacity?

A 2D lattice refers to a two-dimensional arrangement of atoms positioned in a flat plane. In this problem, the lattice is set on the x-y plane. Each atom is coupled with its nearest neighbor via harmonic forces, which are idealized interactions that mimic a perfect spring-like behavior. The dimensions of the lattice are denoted as \(L_x\) and \(L_y\), representing the lengths in the x and y directions respectively.

Given the large number of \(N\) atoms, this lattice resembles a periodic crystal structure in two dimensions. Understanding how waves or vibrations move through this lattice (phonons) is crucial, as these phonons significantly affect the lattice's thermal properties.

Phonon dispersion describes how the frequency of phonons (vibrational quanta) varies with their wavevector \(k\). For a 2D lattice, with atoms interacting through nearest-neighbor harmonic forces, the phonon dispersion relation can be approximated by \(\omega(k) = v_s |k|\). Here, \(\omega(k)\) represents the phonon frequency, \(v_s\) is the speed of sound in the material, and \(k\) is the wavevector.

The dispersion relation essentially tells us how waves of different wavelengths propagate through the lattice. Longer wavelengths (smaller \(k\)) are associated with lower frequencies, while shorter wavelengths (larger \(k\)) possess higher frequencies. This relationship helps in determining the Debye frequency, the maximum possible frequency in the phonon spectrum. Consequently, the Debye frequency for the 2D lattice is given by \(\omega_D = v_s \sqrt{(\pi L_x)^2 + (\pi L_y)^2} \), considering that \(k_{max} = \sqrt{(\pi L_x)^2 + (\pi L_y)^2} \).

Heat capacity is a measure of the amount of heat energy needed to change the temperature of a material. In the context of a 2D lattice at low temperatures (as \(T \to 0\)), the heat capacity behaves differently compared to higher temperatures. Using the Debye model for heat capacity, we account for the contribution of phonons.

At very low temperatures, the heat capacity \(C\) for a 2D lattice is proportional to \(T^2 \). This can be dismissed as

• \(C = A \cdot T^2 \),

where \(A\) is a material-dependent constant. This quadratic dependency arises because only phonons with wavelengths comparable to or longer than the thermal wavelength contribute significantly to the heat capacity. As the temperature increases, more phonons are excited, resulting in higher heat capacity. This relationship is a fundamental aspect of condensed matter physics and aids in predicting how materials behave thermally at low temperatures.